Discover prove: Combining Rational and Irrationalnumbers is 1.2+sqrt2 rational or irrational? Is 1/2*sqrt2 rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf. (a) The sum of rational number r and an irrational number t is irrational. (b) The product of a rational number r and an irrational number t is irrational.

Question
Irrational numbers
asked 2021-03-09
Discover prove: Combining Rational and Irrationalnumbers is \(\displaystyle{1.2}+\sqrt{{2}}\) rational or irrational? Is \(\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}\) rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.

Answers (1)

2021-03-10
Solution:- Combining Rational numbers and irrational numbers:
Rational Numbers:- Numbers which either terminating or none-terminating Repeating. and can be convertible in the form of \(\displaystyle\frac{{p}}{{q}}\) where \(\displaystyle{p},{q}\in{R}\) and \(\displaystyle{q}\ne{0}\) is known as Rational numbers
Irrational numbers:- The Numbers which are non- terminating and non-repeating and all square root of prime numbers is known as Irrational numbers.
Yes \(\displaystyle{12}+\sqrt{{2}}\) is irrational number.
Yes \(\displaystyle{12}\cdot\sqrt{{2}}=\frac{{1}}{\sqrt{{2}}}\) is irrational number.
a) prove that sum of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.To prove:− sum of rational number and irratoional number is irrational.
i.e. (r+t)=irrational
Proof:−It is given that r is rational, t is irrational,and assume that is (r+t) rational.
Since a and a+b are rational, we can write them as fraction. Let, r=ap and (r+t)=bp'
\(\displaystyle\Rightarrow{a}{p}+{t}={b}{p}'\)
\(\displaystyle\Rightarrow{t}={b}{p}'−{a}{p}\)
\(\displaystyle\Rightarrow{t}={b}{p}'+{\left(−{a}{p}\right)}\)
\(\displaystyle\Rightarrow{t}=\) (rational number)+(rational number)
\(\displaystyle\Rightarrow{t}=\) rational number(sum of two rational numbers is always rational number)
but \(\displaystyle{t}=\) irratational which is contradiction.
Hence (r+t)= Irrational number.
Hence proved.
b) prove that product of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.
To prove:−product of rational number and irratoional number is irrational.
i.e. (r*t)=irrational
Proof:−It is given that r is rational, t is irrational, and assume that is (r*t) rational.
Since a and a*b are rational, we can write them as fraction
r=ap
and PSK(r×t)=bp'
\(\displaystyle\Rightarrow{a}{p}\cdot{t}={b}{p}'\)
\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)
\(\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}\)
\(\displaystyle\Rightarrow{t}=\) (rational number)/(rational number)
\(\displaystyle\Rightarrow{t}=\) rational number(rational number of two rational numbers is always rational number)
but t= irratational
which is contradiction.
Hence (r*t)= Irrational number. Hence proved.
0

Relevant Questions

asked 2021-02-11
Consider the following statements. Select all that are always true.
The sum of a rational number and a rational number is rational.
The sum of a rational number and an irrational number is irrational.
The sum of an irrational number and an irrational number is irrational.
The product of a rational number and a rational number is rational.
The product of a rational number and an irrational number is irrational.
The product of an irrational number and an irrational number is irrational.
asked 2020-12-14
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then \(\displaystyle{x}^{{2}}\) is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
asked 2020-12-24
True or False?
1) Let x and y real numbers. If \(\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}\) and \(\displaystyle{x}\ne{y}\), then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.
asked 2021-01-23
The rational numbers are dense in \(\displaystyle\mathbb{R}\). This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in \(\displaystyle\mathbb{R}\) as well.
asked 2020-11-22
How to find a rational number halfway between any two rational numbers given infraction form , add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair.
\(\frac{1}{4}\) and \(\frac{3}{4}\)
asked 2020-11-30
We need to find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair.
\(\frac{1}{100}\) and \(\frac{1}{10}\)
asked 2021-02-01
Given each set of numbers, list the
a) natural Numbers
b) whole numbers
c) integers
d) rational numbers
e) irrational numbers
f) real numbers
\(\displaystyle{\left\lbrace-{6},\sqrt{{23}},{21},{5.62},{0.4},{3}\frac{{2}}{{9}},{0},-\frac{{7}}{{8}},{2.074816}\ldots\right\rbrace}\)
asked 2021-03-12
Find
a) a rational number and
b) a irrational number between the given pair.
\(\displaystyle{3}\frac{{1}}{{7}}\) and \(\displaystyle{3}\frac{{1}}{{6}}\)
asked 2021-02-08
In which set(s) of numbers would you find the number \(\displaystyle\sqrt{{80}}\)
- irrational number
- whole number
- rational number
- integer
- real number
- natural number
asked 2021-02-26
1) Find 100 irrational numbers between 0 and \(\displaystyle\frac{{1}}{{100}}.\)
2) Find 50 rational numbers betwee 1 and 2.
3) Find 50 irrational numbers betwee 1 and 2.
...